Supercalc — Superconducting Properties Calculator

A static, browser-based calculator for superconducting thin-film properties, with a Star Trek: TNG / LCARS theme. No build step, no dependencies, no server-side code — deployable as-is on GitHub Pages.

⚠️ Disclaimer — advisory only, no warranty

Every number Supercalc produces is an approximate estimate from idealized models and literature values that are sample-, deposition-, and measurement-dependent. The results are advisory only and come with no warranty of any kind as to accuracy, completeness, or fitness for any purpose. Do not rely on them as the sole basis for any design, fabrication, procurement, safety, or financial decision. Verify every result against primary references and your own measurements before acting on it. Use is entirely at your own risk. See LICENSE for the full terms.

Panels

1. Surface impedance. Six linked fields: critical temperature Tc, normal state resistivity ρ, sheet resistance R□, film thickness t, sheet inductance L□, and the gap ratio Δ/(k_B·Tc). Enter any three of {Tc, ρ, t, L□} and the rest follow by constraint propagation, with the penetration depth λ and gap energy Δ shown as readouts. The gap ratio defaults to 1.764 (BCS). Leave it blank to solve it from four measured values, and the solved value appears greyed in the empty field.

   R□, L□, and Tc alone fix t only through the excess of L□ over its thin film limit ħR□/(πΔ). That excess is second order in t/λ, so the solver leaves t unsolved rather than amplify rounding. Enter t or ρ as well. An L□ below the thin film limit is rejected. Click a material preset to prefill it. Presets hold ρ, Tc, and the gap, so changing the thickness updates L□.

2. Density of states N₀ / pulse height. Uses the gap and thickness from panel 1 plus a measured single photon response:

   • photon energy or wavelength
   • the inductor's metal area (not its outline), which with t gives the active volume V
   • the loaded Q, f₀, and bath temperature, which must be below Tc/2
   • optionally the pulse decay time τ_qp
   • the phase pulse height θ in degrees, as a loop phase θ₁ or a linearized phase θ₂
   • α_k and η_pb

   It computes the single spin N₀ in eV⁻¹·µm⁻³ and J⁻¹·m⁻³, and, with ρ from panel 1, the diffusion constant D and the dirty limit coherence length ξ. The panel also runs backward, from an entered N₀ to the predicted θ. Choosing a material or clearing panel 1 resets this panel, and the three paper device presets fill it with their resonator and a 2 eV photon. Editing a film property in panel 1 recomputes it.

3. Material explorer. A periodic table of the ambient pressure elemental superconductors. Each clickable cell shows its Tc (gold above 1.3 K, blue below). Click one to prefill panel 1 with its Tc, and with ρ where a low temperature film value is stored.

Physics

The equations follow the Mazin Lab MKIDopt model (mkidopt/materials.py, resonator.py, optimal_filter.py, coordinates.py).

Sheet inductance. With Δ = (gap ratio)·k_B·Tc and ρ the normal state resistivity just above Tc:

λ  = √(ħρ / (πμ₀Δ))  ≈ 105 nm·√(ρ[µΩ·cm] / Tc[K])      Zmuidzinas 2012, Eq. 8
L□ = (μ₀λ/2)·coth(t/2λ)

For t ≪ λ this is ħR□/(πΔ) (Zmuidzinas Eqs. 7, 8, 12). For t ≫ λ it saturates at μ₀λ/2. The coth form is the London slab result for a strip carrying current on both faces, as in MKIDopt. A film over a ground plane, with field on one side only, has the same thin limit but a thick limit of μ₀λ. The thin limit is 15% low for 125 nm Nb (λ ≈ 86 nm) and below 0.3% for the Hf, HfTa, and PtSi presets.

N₀ from a photon pulse (Zmuidzinas Eqs. 64–72 and Gao 2008 thesis, Eqs. 2.96–2.99):

δn_qp = η_pb·E/(ΔV),  δσ₂/σ₂ = −S₂·δn_qp/(2N₀Δ),  δx = (α_k·γ/2)|δσ₂/σ₂|,  θ₂ = 4Q·δx
⇒  N₀ = γ·η_pb·S₂·α_k·Q·E / (Δ²·V·θ₂)

• γ: with λ ∝ σ₂^(−1/2), the coth formula gives γ = −dln L□/dln σ₂ = ½[1 + (t/λ)/sinh(t/λ)]. That is 1 for t ≪ λ and ½ for t ≫ λ (Zmuidzinas Eq. 13). MKIDopt takes γ = 1, within 0.3% for its thin films. For 125 nm Nb it is 0.86. α_k is the fraction of the total inductance contributed by L□.

• θ₂: the exponential amplitude of the pulse in the linearized loop center phase (Zobrist et al. 2021, JATIS 7, 010501).

• Loop phase: a measured loop phase θ₁, with the tone on resonance, linearizes as θ₂ = 2·tan(θ₁/2). At 90° that is 27% above θ₁. This assumes the pulse shifts the frequency without changing the dissipation. Large pulses with a real dissipation response mix the quadratures (Zobrist 2021), so enter a θ₂ linearized from both quadratures for those.

• Pulse peak: an observed pulse peak is h_max·θ₂, where h_max is the maximum of (1 − e^(−t/τ_rise))·e^(−t/τ_qp) and τ_rise = Q/(πf₀). With τ_qp left blank, θ is taken as the amplitude.

• S₂: the thermal Mattis–Bardeen factor, S₂ = 1 + √(2Δ/(πk_BT))·e^(−ξ)·I₀(ξ) with ξ = ħω/(2k_BT) (Zmuidzinas Eq. 72), at the bath temperature. As T → 0 it tends to 1 + √(2Δ/(π²ħω)), not 1. That limit is 2.10 for the PtSi preset at 5.67 GHz.

• η_pb: defaults to 0.59 (Zobrist et al. 2022).

• D and ξ: with N₀ and ρ, D = 1/(2e²N₀ρ) and ξ = √(ħD/Δ).

References:

• J. Zmuidzinas, Superconducting Microresonators: Physics and Applications, Annu. Rev. Condens. Matter Phys. 3, 169 (2012).
• J. Gao, PhD thesis, Caltech (2008).
• N. Zobrist et al., J. Astron. Telesc. Instrum. Syst. 7, 010501 (2021).

Stored data

All material data is stored in js/materials.js (no live fetch).

• Presets:

  • UCSB films from four probe runs: Nb (125 nm), Hf (Tc 503 mK), and HfTa at three Ta powers (Tc 836, 1107, and 2986 mK). Each uses ρ = ρ(300 K)/RRR and the low end of the resistive transition as Tc.
  • From MKIDopt:
    • HfTa with Tc = 0.8 K (unpublished 2026, provisional)
    • PtSi from the MEC array (Zobrist et al. 2019, APL 115, 042601)
    • the In/Hf bilayer (Zobrist et al. 2022, PRL 129, 017701)
    • the UE2 default Hf film

  The MKIDopt TOML configs carry the full provenance of each MKIDopt preset. For the HfTa, PtSi, and In/Hf devices the preset also holds the resonator (loaded Q, metal area of the meander legs, f₀, bath temperature, τ_qp, α_k) and the θ₂ peak that MKIDopt simulates for a 2 eV photon from the measured responsivity η₀. From that pulse the N₀ panel recovers N₀ = η_pb·S₂/(Δ²·η₀): 1.47e47, 8.71e46, and 6.66e46 J⁻¹·m⁻³. MKIDopt's material blocks still hold earlier values for PtSi (1.55e47) and In/Hf (1.23e47, the published supplement value), about 1.8× higher.

• Elements: Tc for the 30 ambient pressure elemental superconductors (Hg excluded), each tagged film or bulk with a source. A resistivity is stored only where a low temperature film value was found (Al, Ru, Ta). A bulk metal's residual resistivity is set by its purity, and a room temperature handbook value overstates the film value by the RRR, so neither belongs in the L□ formula. See the disclaimer above.

Run locally

ES modules require HTTP (not file://):

python3 -m http.server 8000
# open http://localhost:8000

Test

Pure physics modules are unit-tested with Node's built-in runner:

npm test          # → node --test

Deploy (GitHub Pages)

Static site, no build step. Settings → Pages → Deploy from a branch → main → /(root). Pushing to main updates the live site.

License

BSD 3-Clause, with an additional advisory/no-warranty disclaimer for the scientific calculations — see LICENSE. The LCARS theme is an unaffiliated homage.